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Lumifer comments on Median utility rather than mean? - Less Wrong Discussion
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I only brought up Cauchy to show that infinite-variance distributions don't have to be weird and funky. Show a plot of a Cauchy pdf to someone who had, like, one undergrad stats course and she'll say something like "Yes, that's a bell curve" X-/
Actually, there's no need for higher central moments. The mean absolute deviation around the mean (which I would have called the first absolute central moment) bounds the difference between mean and median, and is sharper than the standard deviation.